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Math-2273 · mathematics · Number theory

GPY lower bounds and distances between Gaussian primes

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Can the Goldston--Pintz--Yıldırım method produce an asymptotic count of primes with specified spacing, or at least a lower bound of the correct order? The latter asks for a positive proportion of the admissible tuples under consideration to contain at least two primes. A second question asks whether the method can be adapted to obtain small distances between Gaussian primes.

Formulation references: [S1], [S2].

Short English statement

Can the Goldston--Pintz--Yıldırım method produce an asymptotic count of primes with specified spacing, or at least a lower bound of the correct order? The latter asks for a positive proportion of the admissible tuples under consideration to contain at least two primes. A second question asks whether the method can be adapted to obtain small distances between Gaussian primes.

Sources

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Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2273-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2273

The problem asks for an asymptotic or correct-order GPY count of prime tuples and for an adaptation that produces small distances between Gaussian primes. A bound on the sizes of bounded-step components does not give either count or adaptation: infinitely many close pairs can lie in distinct finite components. The manuscript addresses infinite walks, which are not a clause of this question. Its result is related background rather than a partial solution of the displayed problem.

GPY lower bounds and distances between Gaussian primes

OpenAI's claim

The selected manuscript concerns walks through Gaussian primes with a fixed bound on each step. Fix $D<\infty$. Put an edge between distinct Gaussian primes $z,z'$ when $|z-z'|\le D$. OpenAI asserts a bound $B_D$, depending only on $D$, for the cardinality of each connected component. An infinite path of distinct primes with that step bound is consequently excluded. This is related to Gaussian-prime spacing, but is not a claimed GPY tuple-count estimate or a construction of close prime pairs.

Mathematical statement of the claimed result

Manuscript 1.
For $D<\infty$, form $G_D$ on all Gaussian primes in $\mathbb Z[i]$, joining distinct vertices when their Euclidean distance is at most $D$. The claimed uniform bound is \[ \forall D<\infty\ \exists B_D<\infty\ \forall\text{ connected components }C\subseteq G_D:\quad |C|\le B_D. \] Axis primes are included. Consequently a sequence of distinct Gaussian primes with every successive distance at most $D$ has at most $B_D$ terms; the bound does not depend on its starting point.

Outline of the argument

For manuscript 1 (family 028; 2026-09-26), geometric sampling of a hypothetical long walk produces many residue differences. Shared continuation data create an entropy budget incompatible with avoiding zero residues, yielding a finite periodic sieve barrier and hence the uniform component bound.

Formal verification

For manuscript 1 (family 028; 2026-09-26), OpenAI lists Lean statements for this assertion: for each finite step bound, all connected Gaussian-prime components have a common finite size bound, ruling out infinite injective walks, including those meeting axis primes.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are GaussianMoat.lean.

Their named declarations include OAI.GaussianMoat.fullMain.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

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